Probability tail estimates and convergence results for general distributions using Malliavin calculus and Stein's equations

Richard B Eden, Purdue University

Abstract

For a centered random variable X in a Wiener space, differentiable in the sense of Malliavin, the functional GX is defined using operators from the Malliavin calculus applied to X. By assuming linear or quadratic bounds on GX, we are then able to derive Pearson-type tail bounds on X. Moreover, our results are in fact applicable to comparisons with tails of general distributions, subject to a good control of GX. This is done by relying on a strategy developed by I. Nourdin and G. Peccati, as we relate GX with a Stein equation relative to indicator functions. We apply a similar Nourdin-Peccati analysis to derive bounds on distances between probability laws, this time using a Stein equation relative to Lipschitz functions. We do this by a careful study of the solution to the appropriate Stein equation. With this analysis, we are able to give convergence results to general distributions. Finally, we specialize to convergence results for sequences in a Wiener chaos. We give necessary conditions on the limiting distribution in terms of its cumulants and fractional exponential moments.

Degree

Ph.D.

Advisors

Viens, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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